On structure groups of set-theoretic solutions to the Yang-Baxter equation

This paper explores the structure groups $G_{(X,r)}$ of finite non-degenerate set-theoretic solutions $(X,r)$ to the Yang-Baxter equation. Namely, we construct a finite quotient $\overline{G}_{(X,r)}$ of $G_{(X,r)}$, generalizing the Coxeter-like groups introduced by Dehornoy for involutive solutions. This yields a finitary setting for testing injectivity: if $X$ injects into $G_{(X,r)}$, then it also injects into $\overline{G}_{(X,r)}$. We shrink every solution to an injective one with the same structure group, and compute the rank of the abelianization of $G_{(X,r)}$. We show that multipermutation solutions are the only involutive solutions with diffuse structure group; that only free abelian structure groups are biorderable; and that for the structure group of a self-distributive solution, the following conditions are equivalent: biorderable, left-orderable, abelian, free abelian, torsion free.Comment: 32 pages. Final version. Accepted for publication in Proc. Edinburgh Math. So

Cohomology and extensions of braces

Braces and linear cycle sets are algebraic structures playing a major role in the classification of involutive set-theoretic solutions to the Yang-Baxter equation. This paper introduces two versions of their (co)homology theories. These theories mix the Harrison (co)homology for the abelian group structure and the (co)homology theory for general cycle sets, developed earlier by the authors. Different classes of brace extensions are completely classified in terms of second cohomology groups.Comment: 16 pages. Final version. Accepted for publication in Pacific Journal of Mathematic

Reflection equation as a tool for studying solutions to the Yang-Baxter equation

Given a right-non-degenerate set-theoretic solution $(X,r)$ to the Yang-Baxter equation, we construct a whole family of YBE solutions $r^{(k)}$ on $X$ indexed by its reflections $k$ (i.e., solutions to the reflection equation for $r$). This family includes the original solution and the classical derived solution. All these solutions induce isomorphic actions of the braid group/monoid on $X^n$. The structure monoids of $r$ and $r^{(k)}$ are related by an explicit bijective $1$-cocycle-like map. We thus turn reflections into a tool for studying YBE solutions, rather than a side object of study. In a different direction, we study the reflection equation for non-degenerate involutive YBE solutions, show it to be equivalent to (any of the) three simpler relations, and deduce from the latter systematic ways of constructing new reflections.Comment: 18 pages, 12 figures. Final versio

On Nichols algebras over SL(2,Fq) and GL(2,Fq)

We compute necessary conditions on Yetter-Drinfeld modules over the groups SL(2,Fq) and GL(2,Fq) to generate finite dimensional Nichols algebras. This is a first step towards a classification of pointed Hopf algebras with a group of group-likes isomorphic to one of these groups.Comment: Major exposition revision, including referees remarks. To appear in J. Math. Phys. 13 page

Involutive Yang-Baxter: cabling, decomposability, Dehornoy class

We develop new machinery for producing decomposability tests for involutive solutions to the Yang-Baxter equation. It is based on the seminal decomposability theorem of Rump, and on "cabling" operations on solutions and their effect on the diagonal map. Our machinery yields an elementary proof of a recent decomposability theorem of Camp-More and Sastriques, as well as original decomposability results. It also provides a conceptual interpretation (using the braces language) of the Dehornoy class, a combinatorial invariant naturally appearing in the Garside-theoretic approach to involutive solutions.Comment: 13 pages. Theorem E adde

Exercise prescription to improve clinical practice on cancer patients suffering chemotherapy-induced peripheral neuropathy undergoing treatment: a systematic review

This document aims to summarize and analyze systematically the current body of evidence about the effects of specific exercise proto- cols on physical function, balance control and quality of life in patients with peripheral neuropathy (PNP) induced by chemotherapy. Methods: Systematic Review, Literature survey Specific terms were identified for the literature research in MEDLINE, Scopus, Bandolier, PEDro, and Web of Science. Only studies pub- lished in peer-reviewed journals written in English language were considered. Four manuscripts were classified as eligible with 88 total participants, with an average of 57.1 years old. Quality appraisal classified two studies as high quality investigations while two with low quality. Results were summarized in the following domains: \u201cCIPN symptoms\u201d, \u201cStatic balance control\u201d, \u201cDynamic balance control\u201d, \u201cQuali- ty of life and Physical function\u201d. Results Specific exercise protocols were able to counteract common symptoms of chemotherapy-induced peripheral neuropathy (CIPN) during chemotherapy treatments. Significant improvements were detected on postural control. Additionally, patients\u2019 quality of life and inde- pendence were found ameliorated after exercise sessions, together with reductions on altered sensations and in other peripheral neu- ropathy symptoms. Combined exercise protocols including endurance, strength and sensorimotor training showed larger improvements. Conclusions Exercise prescriptions for cancer patients undergoing chemotherapy with CIPN symptoms should be recommended since these exercise interventions appeared as feasible and have been demonstrated as useful tools to counteract some common side effects of chemother- apeutic agents

Nordic Walking promoted weight loss in overweight and obese people: A systematic review for future exercise prescription

The aim of this systematic review was to analyze the effect of Nordic Walking (NW) on anthropometric parameters, body composition, cardiovascular parameters, aerobic capacity, blood sample, and glucose tolerance in overweight and obese subjects. The main keywords "Nordic Walking" or "Pole Walking", associated with either "obese", "obesity", "overweight", or "weight loss" were used on the online database MEDLINE, PubMed, SPORTDiscus and Scopus. Additionally, references of the studies included were screened to identify eligible articles. Applying the inclusion and exclusion criteria, ten manuscripts were considered as eligible for this review. The results of the studies were categorized in several domains with regard to "anthropometric parameters and body composition", "cardiovascular parameters and aerobic capacity", and "blood sample and glucose tolerance". The results showed positive effects on the anthropometric parameters, body composition, cardiovascular parameters, blood sample, and glucose tolerance. The greatest improvements were observed in supervised and high weekly frequency of NW interventions. NW could be considered as an effective modality through which to involve the obese in physical activity. For weight loss, NW should be prescribed 4-5 times per week, at least 60 min per session, preferably combined with diet control